I honestly think Tessio was just going to hand over the gun, he was a man of honor. He probably had to do the same thing to other people many times in his life, he knew the routine.

I believe it is Cicci who takes the gun from him and it appears to me as there is an understanding between them. Tessio going to hand over the gun, Cicci knows he is just going to hand it over but he still has to take it from him.